Derivative laws and some standard forms of limit

### Derivative Laws

1. **Power Rule**:  
   \[
   \frac{d}{dx}[x^n] = nx^{n-1}
   \]
   For any real number \(n\).

2. **Sum Rule**:  
   \[
   \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]
   \]
   The derivative of a sum is the sum of the derivatives.

3. **Product Rule**:  
   \[
   \frac{d}{dx}[f(x) \cdot g(x)] = f(x) \cdot \frac{d}{dx}[g(x)] + g(x) \cdot \frac{d}{dx}[f(x)]
   \]
   The derivative of a product is the derivative of the first function times the second function plus the first function times the derivative of the second function.

4. **Quotient Rule**:  
   \[
   \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x) \cdot \frac{d}{dx}[f(x)] - f(x) \cdot \frac{d}{dx}[g(x)]}{[g(x)]^2}
   \]
   The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

5. **Chain Rule**:  
   \[
   \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
   \]
   The derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

### Standard Forms of Limits

1. **Limit of a Constant**:  
   \[
   \lim_{{x \to c}} a = a
   \]
   For any constant \(a\), the limit as \(x\) approaches any value \(c\) is \(a\).

2. **Limit of \(x\)**:  
   \[
   \lim_{{x \to c}} x = c
   \]
   As \(x\) approaches \(c\), the limit is \(c\).

3. **Limit of a Power Function**:  
   \[
   \lim_{{x \to c}} x^n = c^n
   \]
   For any positive integer \(n\), as \(x\) approaches \(c\), the limit of \(x^n\) is \(c^n\).

4. **Limit of a Sum**:  
   \[
   \lim_{{x \to c}} [f(x) + g(x)] = \lim_{{x \to c}} f(x) + \lim_{{x \to c}} g(x)
   \]
   The limit of a sum is the sum of the limits.

5. **Limit of a Product**:  
   \[
   \lim_{{x \to c}} [f(x) \cdot g(x)] = \lim_{{x \to c}} f(x) \cdot \lim_{{x \to c}} g(x)
   \]
   The limit of a product is the product of the limits, provided both limits exist.

Would you like more detailed explanations or examples of these rules and limits?  

Here are five questions you might explore further:

1. How does the Chain Rule apply to composite trigonometric functions?
2. What is the derivative of \( e^x \) using the derivative laws?
3. How can the Product Rule be extended to three or more functions?
4. What is the limit of \( \frac{\sin(x)}{x} \) as \( x \) approaches 0, and why?
5. How do you calculate the limit of a function using L'Hôpital's Rule?

**Tip**: When dealing with limits that result in indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), consider using L'Hôpital's Rule to evaluate them effectively.

Explore the derivative laws including Power Rule, Sum Rule, Product Rule, Quotient Rule, and Chain Rule. Learn about standard forms of limits such as constant limits, limits of x, power functions, sums, and products. Suitable for high school and advanced students.

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